This code solves a power system investment planning problem with a time horizon of 15 years. The deterministic version of the problem has 3 decision nodes: one refers to decisions to be taken at present time, one to decisions in 5 years time, and one to decision in 10 years time. The stochastic version is obtained by modeling different possible scenarios for the future of the system in 5 and 10 years. At each node we also compute the cost of operating the system for the following 5 years for given installed capacity. We consider a construction time of 5 years, so new assets installed at present time will only be available in 5 and 10 years, and new capacity installed in 5 years will only be available in 10 years. We model a set of technologies: six thermal units, three storage units, and three renewable generation units.
The stochastic investment planning problem is formulated as
where is the set of decision nodes, each associated with a probability
. The function
gives the cost of operating the system over 5 years. The vector of right-hand side coefficients is given by
here is the accumulated capacity of technology
at node
. Parameters
and
are the relative level of energy demand and the yearly CO2 emission limit, respectively. The vector of cost coefficients
is
where is the uranium fuel price and
the CO2 emission price. Finally, the function
yields the expected total investment and fixed cost, and it is computed as
where the variable is the newly installed capacity of technology
at node
. Parameters
and
are the unitary investment and fixed cost of technology
at node
. Parameters
. The accumulated capacity
at node
is computed as the sum of the historical capacity and the newly installed capacity
in all nodes
ancestors to
.
We consider three possible sources of uncertainty, i.e., ,
, and
. Each uncertain parameters has 3 possible outcomes in 5 years, each of which is linked to 3 additional possible outcomes in 10 years. The result is 9 possible trajectories for each uncertainty, all with the same probability. We consider 4 different cases of the investment problem. Case 0 is the deterministic version, where
,
, and
are deterministic parameters (weighted average of the scenarios). Then, case 1 has 1 uncertain parameter, case 2 has 2 uncertain parameters, and case 3 has 3 uncertain parameters. The number of decision nodes for the 4 versions of the problem is
| case | uncertain parameters | decision nodes |
|---|---|---|
| 0 | - | 3 |
| 1 | 13 | |
| 2 | 91 | |
| 3 | 757 |
The investment problem can be solved with two algorithms: Algorithm 1 (Stand_Bend) and Algorithm 2 (Adapt_Bend). Both Algorithm Stand_Bend and Adapt_Bend are presented in ''Benders Decomposition with Adaptive Oracles for Large Scale Optimization'' (http://www.optimization-online.org/DB_HTML/2019/08/7327.html).
Install Julia 1.4 (with packages JuMP.jl 0.21, Gurobi.jl, Suppressor.jl, CSV.jl, JLD.jl, Printf.jl, Clustering.jl, ParallelDataTransfer.jl) and Gurobi 9.0.
Open the terminal and change directory to the project folder.
bash$ cd ~/path_to_folder/Stand_and_Adapt_Bend/serialStart Julia and include "main.jl"
bash$ julia
_
_ _ _(_)_ | Documentation: https://docs.julialang.org
(_) | (_) (_) |
_ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help.
| | | | | | |/ _` | |
| | |_| | | | (_| | | Version 1.4.0 (2020-03-21)
_/ |\__'_|_|_|\__'_| | Official https://julialang.org/ release
|__/ |
julia> include("main.jl")You will be asked to select the case study (0, 1, 2, or 3),
case 0 -> 0 uncertain parameters
case 1 -> 1 uncertain parameters
case 2 -> 2 uncertain parameters
case 3 -> 3 uncertain parameters
select case study:
1 and the algorithm (0, 1, 2, or 3) used to solve the problem
algorithm 0 -> deterministic equivalent
algorithm 1 -> Stand_Bend (Standard Benders)
algorithm 2 -> Adapt_Bend (Benders with Adaptive Oracles)
algorithm 3 -> Zaker_Bend (Benders, Zakeri et al.)
select Benders-type algorithm:
2 if algorithm Adapt_Bend is selected, input the number w of subproblems to solve at each iteration
number of subproblems to solve each iter j:
input w (0 < w < 12)
1
if algorithm Zaker_Bend is selected, input the coefficient q to update the optimality tolerance (10 suggested)
select parameter σ:
input σ (1 < σ <= 100)
10
The algorithm starts and stops once reached the predifined tolerance, e.g.,
*/--------------------------------/*
algorithm : adapt_bend
case : 1
subs per iter : 1
investment nodes : 4
operational nodes : 12
*/--------------------------------/*
k = 1, δ = 99.791 %, t = 2.31 s
k = 2, δ = 99.542 %, t = 0.93 s
...
k = 69, δ = 0.010 %, t = 2.65 s
k = 70, δ = 0.009 %, t = 4.24 s
*/--------------------------------/*
*/--------------------------------------------------------------------/*
co2 emission limit : uncertain
co2 emission cost : known
uranium cost : known
investment nodes : 4
operational nodes : 12
optimal objective : 1.381 x 10^11 £
*/--------------------------------------------------------------------/*
optimal investments (GW) @ 0 years
----------------
tech. i1
----------------
coal 0.0
coalccs 0.0
OCGT 0.0
CCGT 13.3
diesel 1.8
nuclear 0.0
pumpL 0.0
pumpH 0.0
lithium 0.0
onwind 19.0
offwind 0.0
solar 0.0
----------------
optimal investments (GW) @ 5 years
------------------------------
tech. i1 i2 i3
------------------------------
coal 0.0 0.0 0.0
coalccs 0.0 0.0 2.3
OCGT 0.0 0.0 0.0
CCGT 4.4 0.9 0.0
diesel 2.5 2.5 0.1
nuclear 5.4 8.6 10.0
pumpL 0.0 0.0 0.0
pumpH 0.0 0.0 0.0
lithium 0.0 0.0 0.0
onwind 0.0 0.0 0.0
offwind 0.0 0.0 0.0
solar 0.0 0.0 0.0
------------------------------
*/--------------------------------------------------------------------/*
computational results:
-------------------------------------------------------------------
ϵ-target (%) | iters time (s) RMP (%) SP (%) Oracles (%)
-------------------------------------------------------------------
1.00 | 44 87 0.10 99.46 0.11
0.10 | 51 106 0.09 99.52 0.12
0.01 | 70 155 0.07 99.58 0.13
-------------------------------------------------------------------
*/--------------------------------------------------------------------/*Open the terminal and change directory to the project folder.
bash$ cd ~/path_to_folder/Stand_and_Adapt_Bend/parallelModify file /functions/load_cluster.jl if loading workers on multiple machines, specify the ip address of the machine, the path to the gurobi license, and the path to the julia executable.
machine_1 = Dict("hostip" => "000.000.000.000",
"grblcs" => "path_to_gurobi_license",
"jlpath" => "path_to_julia_exec")
Start Julia and include "main.jl"
bash$ julia
_
_ _ _(_)_ | Documentation: https://docs.julialang.org
(_) | (_) (_) |
_ _ _| |_ __ _ | Type "?" for help, "]?" for Pkg help.
| | | | | | |/ _` | |
| | |_| | | | (_| | | Version 1.4.0 (2020-03-21)
_/ |\__'_|_|_|\__'_| | Official https://julialang.org/ release
|__/ |
julia> include("main.jl")You will be asked to select the case study (0, 1, 2, or 3),
case 0 -> 0 uncertain parameters
case 1 -> 1 uncertain parameters
case 2 -> 2 uncertain parameters
case 3 -> 3 uncertain parameters
select case study:
2 select the algorithm (1 or 2) used to solve the problem
algorithm 1 -> Stand_Bend (Standard Benders)
algorithm 2 -> Adapt_Bend (Benders with Adaptive Oracles)
select Benders-type algorithm:
2 and the number of workers to start up on every computer
select number of workers on machine0 (local_machine)
select 0 <= w <= 11 :
2
select number of workers on machine1 (000.000.000.000)
select 0 <= w <= 9 :
0 The algorithm starts and stops once reached the predifined tolerance, e.g.,
*/--------------------------------/*
algorithm : adapt_bend
case : 2
investment nodes : 10
operational nodes : 90
workers : 2
*/--------------------------------/*
k = 1, δ = 99.759 %, t = 3.27 s
k = 2, δ = 99.116 %, t = 2.23 s
...
k = 144, δ = 0.013 %, t = 3.76 s
k = 145, δ = 0.009 %, t = 3.0 s
*/--------------------------------/*
*/--------------------------------------------------------------------/*
co2 emission limit : uncertain
co2 emission cost : known
uranium cost : known
investment nodes : 10
operational nodes : 90
optimal objective : 1.374 x 10^11 £
*/--------------------------------------------------------------------/*
optimal investments (GW) @ 0 years
----------------
tech. i1
----------------
coal 0.0
coalccs 1.0
OCGT 0.0
CCGT 11.5
diesel 2.0
nuclear 0.0
pumpL 0.0
pumpH 0.0
lithium 0.0
onwind 19.0
offwind 0.0
solar 0.0
----------------
optimal investments (GW) @ 5 years
------------------------------------------------------------------------
tech. i1 i2 i3 i4 i5 i6 i7 i8 i9
------------------------------------------------------------------------
coal 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
coalccs 6.5 6.5 6.5 0.0 0.0 1.4 0.0 0.0 1.4
OCGT 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
CCGT 0.0 0.0 0.0 6.0 2.9 0.0 5.6 2.7 0.0
diesel 0.0 0.0 0.0 2.1 2.1 1.3 2.6 2.4 1.4
nuclear 10.0 10.0 10.0 4.8 7.9 10.0 4.7 7.8 10.0
pumpL 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
pumpH 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
lithium 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
onwind 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
offwind 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
solar 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
------------------------------------------------------------------------
*/--------------------------------------------------------------------/*
computational results:
-------------------------------------------------------------------
ϵ-target (%) | iters time (s) RMP (%) SP (%) Oracles (%)
-------------------------------------------------------------------
1.00 | 38 112 3.17 95.89 0.65
0.10 | 83 263 3.87 94.99 0.96
0.01 | 145 490 4.77 93.76 1.33
-------------------------------------------------------------------
*/--------------------------------------------------------------------/*